Stability of supermartingale optimal transport problems

Stabilit y of sup ermartingale optimal transp ort problems Sh uo qing Deng ∗ Gao yue Guo † Domin yk as Norgilas ‡ Abstract W e in vestigate stabilit y prop erties of w eak sup ermartingale optimal transport (WSOT) problems on R . F or probability measures µ, ν P P r satisfying µ ď cd ν (equiv alen tly , Π S p µ, ν q ‰ H ), w e consider sup ermartingale couplings π “ µ p d x q π x p d y q and the weak transp ort functional V C S p µ, ν q : “ inf π P Π S p µ,ν q ż R C p x, π x q µ p d x q , for some appropriate cost function C : R ˆ P r Ñ R . Our first main contribution is an approximation result in adapted W asserstein distance: under W r -con vergence of marginals p µ k , ν k q Ñ p µ, ν q with µ k ď cd ν k , any π P Π S p µ, ν q can b e appro ximated by π k P Π S p µ k , ν k q suc h that A W r p π k , π q Ñ 0. As a consequence, we obtain the con tinuit y of the functional p µ, ν q ÞÑ V C S p µ, ν q , and the monotonicity principle for WSOT. Con ten ts 1 In tro duction 2 1.1 Main con tribution: stabilit y and monotonicity principle . . . . . . . . . . . . 3 1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Main results 5 2.1 Appro ximation in adapted W asserstein top ology . . . . . . . . . . . . . . . . 5 2.2 Stabilit y of the WSOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Sup ermartingale C -monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Pro ofs of Theorems 2.3 and 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4.1 Pro ofs of Theorem 2.3: stability of the v alue function . . . . . . . . . 7 2.4.2 Pro of of Theorem 2.6: monotonicity principle . . . . . . . . . . . . . . 9 3 Pro of of Theorem 2.1 11 3.1 Outline of the pro of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1.1 Reduction to the strict sup ermartingale part (to the right of x ˚ ) . . . 12 3.1.2 Useful lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.3 Lo calisation and barycen tre correction . . . . . . . . . . . . . . . . . . 14 ∗ The Hong Kong Universit y of Science and T ec hnology , Departmen t of Mathematics (mas- deng@ust.hk). S. Deng is supported b y the Hong Kong Univ ersity of Science and T echnology Start-up Gran t No. R9826 and Hong Kong RGC Early Career Sc heme (ECS) Grant No. 26307125. † Univ ersit´ e Paris-Sacla y Cen traleSup´ elec, Lab oratoire MICS and CNRS FR-3487 (gao yue.guo@centralesupelec.fr). Guo ackno wledges financial supp ort b y the gran t ANR JCJC MA TH-SP A. ‡ North Carolina State Univ ersity , Department of Mathematics (dnorgil@ncsu.edu). 1 3.1.4 Completion and gluing . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1.5 Final adjustmen t of the second marginal . . . . . . . . . . . . . . . . . 16 3.2 Pro ofs of the relev ant results . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2.1 Pro of of the results in Section 3.1.1 . . . . . . . . . . . . . . . . . . . . 17 3.2.2 Pro of of the results in Section 3.1.2 . . . . . . . . . . . . . . . . . . . . 17 3.2.3 Pro of of the results in Section 3.1.3 . . . . . . . . . . . . . . . . . . . . 19 3.2.4 Pro of of the results in Section 3.1.4 . . . . . . . . . . . . . . . . . . . . 22 4 App endix 25 4.1 Irreducible decomp osition for supermartingale couplings . . . . . . . . . . . . 25 4.2 A lo calisation lemma in adapted W asserstein top ology . . . . . . . . . . . . . 25 4.3 An alternativ e form of WSOT . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.4 Comp etitor and finite optimalit y . . . . . . . . . . . . . . . . . . . . . . . . . 27 1 In tro duction Ov er the past tw o decades, c onstr aine d optimal tr ansp ort —optimal mass transfer sub ject to additional structural constrain ts—has b ecome a cen tral topic. A prominent example is martingale optimal tr ansp ort (MOT), motiv ated by model-indep endent pricing and hedging in mathematical finance. Let X , Y b e Polish subspaces of R and µ P P 1 p X q , ν P P 1 p Y q and a coupling π p dx, dy q “ µ p dx q π x p dy q P Π p µ, ν q , the martingale and sup ermartingale constrain ts are imp osed on the conditional barycentres: Π M p µ, ν q : “ ! π P Π p µ, ν q : ż Y y π x p dy q “ x for µ -a.e. x ) , (1) Π S p µ, ν q : “ ! π P Π p µ, ν q : ż Y y π x p dy q ď x for µ -a.e. x ) . (2) W e refer to [ 11 ] for the discrete-time form ulation and to [ 21 ] for the con tinuous-time formulation. A basic feasibilit y question concerns the non-emptiness of these constrain t sets. By the Strassen-t yp e theorems (Strassen [35]), this is characterized b y sto c hastic orders: Π M p µ, ν q ‰ H ð ñ µ ď c ν, Π S p µ, ν q ‰ H ð ñ µ ď cd ν, where ď c and ď cd denote the c onvex or der and the de cr e asing c onvex or der , resp ectiv ely . W eak optimal transp ort (W OT) extends the classical transp ort problem b y allowing the cost to dep end on the conditional law rather than only on the lo cations of transp orted p oin t. Giv en a measurable cost function C : X ˆ P p Y q Ñ R , the weak optimal transp ort problem is defined as: V p µ, ν q : “ inf π P Π p µ,ν q ż X C p x, π x q µ p dx q . (3) This form ulation go es bac k to the works of Marton [ 31 , 30 ] and T alagrand [ 36 , 37 ], and has since b een dev elop ed extensiv ely; see, for example, [1, 20, 22, 34, 33, 24, 23, 2, 4, 3, 6, 5]. In this pap er w e study we ak sup ermartingale optimal tr ansp ort (WSOT) on X ˆ P p Y q , namely V C S p µ, ν q : “ inf π P Π S p µ,ν q ż Y C p x, π x q µ p dx q , (4) 2 whic h extends b oth classical sup ermartingale transp ort and weak martingale transp ort by allo wing costs that dep end on the full conditional law. 1.1 Main con tribution: stabilit y and monotonicit y principle A central issue for both theory and applications is stability with resp ect to the marginals: giv en p µ k , ν k q Ñ p µ, ν q , do es one ha ve V C S p µ k , ν k q Ñ V C S p µ, ν q , and, more delicately , can the feasible sets Π S p µ k , ν k q appro ximate Π S p µ, ν q in a top ology that is fine enough to control the conditional la ws? In the martingale case, stabilit y for w eak costs was initiated by Juillet [ 28 ] and Guo and Ob l´ oj [ 25 ], and later established in full generalit y on the line by Beiglb¨ oc k, Jourdain, Margheriti and Pammer [ 13 , 12 ] using the adapted W asserstein top ology . In higher dimensions, stabilit y may fail ev en for martingale optimal transp ort [16]. F or sup ermartingale transp ort, contin uity of the v alue has been obtained for certain canonical constructions, notably via shadow measures [ 7 , 8 ]. These results are quan titative but rely on sp ecific couplings. The present work dev elops a general stability theory for w eak sup ermartingale optimal transport on the line. F or the monotonicity principle in MOT, Beiglb¨ oc k and Juillet [ 14 ] established the cyclical monotonicit y principle in the martingale setting and introduced the left-curtain coupling, whic h is optimal for a class of cost functions. Motiv ated b y ideas from MOT, Beiglb¨ oc k, Co x and Huesmann [ 9 ] dev elop ed a geometric c haracterization of Sk orokho d em b eddings with certain optimality prop erties, leading to a systematic construction of optimal em beddings. F urther studies on monotonicit y principles can b e found in Guo, T an and T ouzi [27, 26]. • Assume that µ k ď cd ν k for all k and that p µ k , ν k q Ñ p µ, ν q in W 1 . Our first main result (Theorem 2.1) is an appro ximation theorem in adapted W asserstein distance: ev ery π P Π S p µ, ν q can b e appro ximated by couplings π k P Π S p µ k , ν k q suc h that A W 1 p π k , π q Ñ 0. This extends [ 12 , Theorem 2.6] from martingales to sup ermartingales and requires new argumen ts to handle the completion of sub-probabilit y measures under the decreasing con vex order. As a consequence, w e obtain stability of the WSOT problem for a suitable class of costs, including contin uit y of the optimal v alue and conv ergence of optimizers. • Our second main result is a monotonicit y principle for supermartingale optimal transp ort, in b oth w eak and classical formulations. It provides a geometric c haracterization of the supp ort of optimal couplings and extends the martingale monotonicit y principle of [ 13 ] to the sup ermartingale setting. Organization of the pap er. Section 1.2 introduces notation and preliminary results. Section 2 con tains the main theorems on appro ximation, stabilit y , and monotonicity . Section 3 is dev oted to the pro of of the approximation theorem, and Section 4 contains the remaining pro ofs. 1.2 Preliminaries This section fixes notation and recalls results used throughout the pap er. Unless stated otherwise, w e work on R . 3 Measures, couplings and disin tegration. F or a Polish space X , let M p X q denote the set of finite, p ositiv e Borel measures on X , and P p X q Ă M p X q denote the set of probability measures. F or r ě 1, let M r p X q and P r p X q b e the subsets of measures with finite r th moment. W e write M : “ M p R q , P : “ P p R q and similarly M r , P r . F or η P M 1 , define its first moment and barycen tre by η : “ ż R x η p dx q , ba ry p η q : “ η η p R q if η p R q ą 0 . W e denote by supp p η q the closed supp ort of η , by I η the smallest interv al containing supp p η q , and b y ℓ η , r η the endp oin ts of I η (p ossibly infinite). F or µ P M p X q and ν P M p Y q with equal mass, Π p µ, ν q denotes the set of couplings on X ˆ Y . W asserstein distance. F or µ, ν P M r p R q with µ p R q “ ν p R q , denote by W r p µ, ν q their W asserstein distance of order r . F or r “ 1, the follo wing duality holds, i.e., W 1 p µ, ν q “ sup } f } Lip ď 1 ! ż f dµ ´ ż f dν ) . (5) F or η P M p R q , define its distribution function F η p x q : “ η pp´8 , x sq and the left-contin uous quan tile F ´ 1 η p u q : “ inf t x P R : F η p x q ě u u , u P p 0 , η p R qq . Then the quantile formula yields W 1 p µ, ν q “ ż µ p R q 0 ˇ ˇ F ´ 1 µ p u q ´ F ´ 1 ν p u q ˇ ˇ du “ ż R | F µ p x q ´ F ν p x q| dx. (6) Adapted W asserstein dista nce. Let π “ µ ˆ π x P M r p R 2 q and π 1 “ µ 1 ˆ π 1 x 1 P M r p R 2 q with µ p R q “ µ 1 p R q . The adapted W asserstein distance is defined b y A W r p π , π 1 q : “ ˆ inf γ P Π p µ,µ 1 q ż R 2 ´ | x ´ x 1 | r ` W r p π x , π 1 x 1 q r ¯ γ p dx, dx 1 q ˙ 1 { r . (7) F or r “ 1, with the canonical em b edding J p π q : “ µ p dx q δ π x p dp q P M 1 ` R ˆ P 1 ˘ , one has AW 1 p π , π 1 q “ W 1 p J p π q , J p π 1 qq . Put function and conv ergence in W 1 . F or η P M 1 , define the put and p oten tial functions P η , U η : R Ñ R ` b y P η p x q : “ ż R p x ´ t q ` η p dt q , U η p x q : “ ż R | x ´ t | η p dt q . These functions are conv ex and determine η via second deriv ativ es in the sense of distributions; see [18, 19, 29]. Lemma 1.1. L et p µ k q k ě 1 Ă P 1 p R q and µ P P 1 p R q . Then W 1 p µ k , µ q Ñ 0 ð ñ P µ k Ñ P µ p ointwise on R and bary p µ k q Ñ bary p µ q , and e quivalently with uniform c onver genc e of P µ k on R . In p articular, sup x P R ˇ ˇ P µ k p x q ´ P µ p x q ˇ ˇ ď W 1 p µ k , µ q . 4 Sto c hastic orders and (sup er)martingale couplings. F or η , χ P M , we recall: • η ď χ if ş f dη ď ş f dχ for all non-negative and b ounded f ; • η ď c χ if η , χ P M 1 , η p R q “ χ p R q and ş f dη ď ş f dχ for all conv ex f ; • η ď cd χ if η , χ P M 1 , η p R q “ χ p R q and ş f dη ď ş f dχ for all conv ex nonincreasing f . F or measures of equal mass, one has the characterization η ď cd χ ð ñ P η ď P χ on R , and if additionally η “ χ , then η ď c χ . F urther, given µ, ν P M 1 with µ p R q “ ν p R q , recall that Π M p µ, ν q Ă Π p µ, ν q (resp. Π S p µ, ν q Ă Π p µ, ν q ) denotes the collection of martingale (resp. sup ermartingale) couplings betw een µ and ν . Theorem 1.2 (Strassen-type feasibility) . L et µ, ν P M 1 with µ p R q “ ν p R q . Then Π S p µ, ν q ‰ H ð ñ µ ď cd ν, Π M p µ, ν q ‰ H ð ñ µ ď c ν. 2 Main results W e state the three main results in this section: first, an appro ximation theorem for sup ermartin- gale couplings in the adapted W asserstein top ology; second, contin uit y of the corresp onding w eak sup ermartingale optimal transp ort problem; and finally , a monotonicity principle for WSOT and classical SOT. 2.1 Appro ximation in adapted W asserstein top ology Throughout, let p µ k , ν k q k ě 1 Ă P r ˆ P r and p µ, ν q P P r ˆ P r satisfy W r p µ k , µ q ` W r p ν k , ν q Ñ 0 and µ k ď cd ν k for all k ě 1 . Without an y risk of confusion, w e write equally that p µ k , ν k q k ě 1 con verge to p µ, ν q in W r or p µ k , ν k q Ñ p µ, ν q . Then it holds µ ď cd ν b y Lemma 1.1. Theorem 2.1 (Appro ximation of sup ermartingale couplings) . L et p µ k , ν k q k ě 1 c onver ge to p µ, ν q in W r and assume µ k ď cd ν k for al l k ě 1 . Then for every π P Π S p µ, ν q , ther e exists a se quenc e π k P Π S p µ k , ν k q such that A W r p π k , π q Ý Ñ 0 , k Ñ 8 . Remark 2.2 (Con text and nov elty) . Theorem 2.1 is the sup ermartingale analogue of the martingale appro ximation theorem of [ 12 , Theorem 2.6]. In the martingale case, one can w ork on the irreducible decomp osition and glue approximations without losing the barycentre constrain t. F or sup ermartingales, an additional difficulty is that the constraint is inequalit y- v alued; in particular, the gluing step requires a careful barycentre adjustmen t using free mass to the left of the compact region. More precisely , when completing the sub-probability measures ˆ µ k and ˜ ν k (see Section 3.1.4) to reco ver probability measures with the prescrib ed marginals, one needs to prov e that the put potential of the residual part of µ k is dominated b y that of the residual part of the second marginal on the whole real line. The argument is decomposed in to the comparison on an in terv al J and on its complemen t J c . On J the pro of is straightforw ard. On J c , unlike in the martingale case of [ 12 ], b oth sides of the comparison are nontrivial, so one must return to the definitions of the auxiliary measures and establish the comparison directly . 5 2.2 Stabilit y of the WSOT Theorem 2.3. Fix r P r 1 , 8q , and let X and Y b e Polish subsp ac es of R . L et C : X ˆ P r p Y q Ñ R b e me asur able, c ontinuous and c onvex in the se c ond ar gument. Assume that ther e exists K ą 0 such that | C p x, m q | ď K ˆ 1 ` | x | r ` ż Y | y | r m p dy q ˙ , @p x, m q P X ˆ P r p Y q . Fix µ P P r p X q , ν P P r p Y q with µ ď cd ν . L et µ k P P r p X q , ν k P P r p Y q satisfy µ k ď cd ν k , and let µ k , ν k c onver ge to µ and ν under W r , r esp e ctively. Then lim k Ñ8 V C S p µ k , ν k q “ V C S p µ, ν q (8) holds under one of the fol lowing c onditions: (A”) C is c ontinuous; (B”) µ k c onver ges str ongly to µ . F or k P N , let π ˚ ,k P Π S p µ k , ν k q b e a minimiser of V C S p µ k , ν k q . Then any ac cumulation p oint of p π ˚ ,k q k P N (with r esp e ct to the we ak c onver genc e) is a minimiser of V C S p µ, ν q . If V C S p µ, ν q has a unique minimiser π ˚ P Π S p µ, ν q , then π ˚ ,k k Ñ8 Ý Ý Ý Ñ π ˚ in W r , (9) If mor e over, C is strictly c onvex in the se c ond ar gument, then the c onver genc e in (9) holds in A W r . Pr o of. Com bining Prop osition 2.9 and Prop osition 2.10, w e draw the conclusion. 2.3 Sup ermartingale C -monotonicity A by-product of the SOT stability result, is to pro ve that supermartingale C -monotonicit y is a necessary and sufficient criterion for optimalit y: π ˚ P Π S p µ, ν q minimises the WSOT problem if and only if µ p dx q δ π ˚ x p dp q is concentrated on a sup ermartingale C -monotone set. Definition 2.4 (Sup ermartingale C -monotonicit y .) . The Borel set Γ Ă R ˆ P 1 is sup er- martingale C -monotone if and only if there exists M 0 , M 1 Ă R , such that for an y N P N , any collection p x 1 , p 1 q , ¨ ¨ ¨ , p x N , p N q P Γ and q 1 , ¨ ¨ ¨ q N P P 1 suc h that ř N i “ 1 p i “ ř N i “ 1 q i and ż R y p i p dy q “ ż R y q i p dy q , @ i P I 1 ; ż R y p i p dy q ě ż R y q i p dy q , @ i P I 0 , w e hav e N ÿ i “ 1 C p x i , p i q ď N ÿ i “ 1 C p x i , q i q . 6 where, I 0 “ t i : x i P M 0 u , I 1 “ t i : x i P M 1 u . In the follo wing, we say a coupling π P Π S p µ, ν q is Sup ermartingale C -monotone if there exists a Sup ermartingale C -monotone set Γ with p x, π x q P Γ for µ p dx q -almost every x, with M 0 , M 1 defined as the martingale region of π to the left and righ t of x ˚ (recall that x ˚ is defined in Section 4.1) In addition, a probabilit y measure P P P p R ˆ P 1 , whic h is supp orted on a martingale C -monotone set, is called sup ermartingale C -monotone. Remark 2.5. Notice that for the sup ermartingale C -monotonicit y of the supp ort set Γ, w e don’t require ş R y p i p dy q “ ş R y q i p dy q , for i P I 1 , which is consisten t with [ 32 ]. W e then c ho ose M 0 and M 1 sp ecifically as in the definition of sup ermartingale C -monotonicit y for coupling π . W e emphasize here that the region of M 1 dep ends only on the marginals, while M 0 dep ends on the sp ecific coupling. In the follo wing, w e shall prov e that the ab o ve sup ermartingale C -monotonicit y is a necessary and sufficient condition for the optimalit y of the optimizer. W e ha ve the following main theorem: Theorem 2.6. L et µ ď cd ν P P r p R q with r ě 1 . Assume that C : R ˆ P r p R q Ñ R is me asur able, c ontinuous and c onvex in the se c ond ar gument and ther e exists K ą 0 so that @p x, p q P R ˆ P r p R q , | C p x, p q| ď K ˆ 1 ` | x | r ` ż R | y | r p p dy q ˙ . Then π P Π S p µ, ν q is sup ermartingale C -monotone w.r.t. some Γ if and only if π is optimal for (WSOT). Pr o of. Com bine Prop osition 2.12 and Prop osition 2.14. Using the ab o ve main theorem, we can obtain the m onotonicit y principle of classical SOT. Recall that the definition of finite optimality w as given in 4.4. Corollary 2.7. (Monotonicity principle for SOT). L et r ě 1 , µ ď cd ν P P r , c : R ˆ R Ñ R b e me asur able and such that y ÞÑ c p x, y q is c ontinuous for al l x P R and sup p x,y qP R 2 | c p x,y q| | 1 `| x | r `| y | r ă 8 . Then π P Π S p µ, ν q is optimal for (SOT) if and only if it is c onc entr ate d on a finitely optimal set. Pr o of. The argument is similar as [13, Corollary 3.4], and we omit it here. 2.4 Pro ofs of Theorems 2.3 and 2.6 2.4.1 Pro ofs of Theorem 2.3: stabilit y of the v alue function Remark 2.8. Let r ě 1 and X , Y b e P olish subspace of R . F or a measurable C : X ˆ P r p Y q Ñ R Y t8u , consider ˜ C : X ˆ P r p Y q Ñ R Y t8u defined by ˜ C p x, m q “ C p x, m q I t ş Y y m p dy qď x u ` 8 I t ş Y y m p dy qą x u . If C is conv ex in its second argumen t (resp ectiv ely is low er semicontin uous in either its second argument or in b oth arguments, resp ectively satisfies C p x, m q ě ´ K ` 1 ` | x | r ` ş Y | y | r m p dy q ˘ ), then so do es ˜ C . Indeed, the identit y map R Q y Ñ y P R b elongs to Φ r p R q “ t h : R Ñ R : h is contin uous and D α ą 0 , @ x P R , | h p x q | ď α p 1 ` | x | r u , and th us tp x, m q P X ˆ P r p Y q : ş Y y m p dy q ď x u Ă X ˆ P r p Y q is closed. Also, it is easy to see that, for a fixed x P R , t m P P r p Y q : x “ ş Y y m p dy qu is con vex. 7 The pro of of Theorem 2.3 is fulfilled b y the following t wo prop ositions. Prop osition 2.9 (Attainmen t and low er semicon tinuit y) . Fix r P r 1 , 8q and let X , Y b e Polish subsp ac e of R . Assume that C : X ˆ P r p Y q Ñ R is me asur able, c onvex and lower semic ontinous in the se c ond ar gument, and ther e exists K ą 0 so that C p x, m q ě ´ K ˆ 1 ` | x | r ` ż Y | y | r m p dy q ˙ , @p x, m q P X ˆ P r p Y q . F or µ P P r p X q , ν P P r p Y q with µ ď cd ν , ther e exists a minimizer π ˚ P Π S p µ, ν q for V C S p µ, ν q . If C is strictly c onvex in the se c ond ar gument and V C S p µ, ν q is finite, then the minimizer is unique. F or k P N , let µ k P P r p X q , ν k P P r p Y q with µ k ď cd ν k c onver ge in W r to µ and ν , r esp e ctively. If one of the fol lowing holds: (A) C is lower semic ontinuous in b oth ar guments; (B) µ k c onver ges str ongly to µ . Then V C S p µ, ν q ď lim inf k Ñ8 V C S p µ k , ν k q . Pr o of. Note that V C S p µ, ν q “ inf π P Π S p µ,ν q ż X C p x, π x q µ p dx q “ inf π P Π p µ,ν q ż X ˜ C p x, π x q µ p dx q “ V ˜ C p µ, ν q , where ˜ C : X ˆ P r p Y q Ñ R is defined by ˜ C p x, m q “ C p x, m q I t ş Y y m p dy qď x u ` 8 I t ş Y y m p dy qą x u ; recall Remark 2.8. Then, b y following verbatim the pro of of [ 13 , Theorem 2.4], one obtains the statemen ts concerning V ˜ C (and th us also V C S ). Prop osition 2.10 (Upper semicon tinuit y) . Fix r P r 1 , 8q and let X , Y b e Polish subsp ac e of R . Assume that C : X ˆ P r p Y q Ñ R is me asur able, upp er semic ontinous in the se c ond ar gument, and ther e exists K ą 0 so that C p x, m q ď K ˆ 1 ` | x | r ` ż Y | y | r m p dy q ˙ , @p x, m q P X ˆ P r p Y q . F or k P N , let µ k P P r p X q , ν k P P r p Y q with µ k ď cd ν k c onver ge in W r to µ and ν , r esp e ctively. If one of the fol lowing holds: (A’) C is upp er semic ontinuous in b oth ar guments; (B’) µ k c onver ges str ongly to µ . Then lim sup k Ñ8 V C S p µ k , ν k q ď V C S p µ, ν q . Pr o of. Fix π P Π S p µ, ν q . Then our appro ximation result, Theorem 2.1, ensures that there exists a sequence π k P Π S p µ k , ν k q , k P N , which con v erges to π in A W r . Then the result follo ws from the Portman teau-type arguments; see [ 13 , Section A.2]. In particular, giv en the appro ximation result, we can follo w verbatim the pro of of [13, Theorem 2.8]. 8 2.4.2 Pro of of Theorem 2.6: monotonicit y principle W e no w pro ceed to prov e the monotonicity principle for (WSOT). W e will first use the stability result to pro ve the sufficiency side, and then use an adapted v ersion of an abstract result in [10] to prov e the necessary side. In order to prov e the sufficiency , we first argue the following sufficiency for finite optimality . Lemma 2.11. If π P Π S p µ, ν q is a finitely supp orte d c oupling of the form 1 N ř N i “ 1 δ p x i q p dx q p i p dy q for x 1 ă ¨ ¨ ¨ ă x n P R and p 1 , ¨ ¨ ¨ , p n P P 1 p R q , and is sup ermartingale C -monotone w.r.t. some Γ , then it is optimal. Pr o of. Let any martingale coupling π 1 P Π S p µ, ν q of the form 1 N ř N i “ 1 δ p x i q p dx q q i p dy q for x 1 ă ¨ ¨ ¨ ă x n P R and q 1 , ¨ ¨ ¨ , q n P P 1 p R q , and suc h that ř N i “ 1 p i “ ř N i “ 1 q i . No w define M 0 , M 1 as the martingale region of π to the left and righ t of x ˚ . It is clear that @ i P t 1 , . . . , N u , ş R y p i p dy q “ x i “ ş R y q i p dy q for all x i P M 1 and @ i P t 1 , . . . , N u , ş R y p i p dy q “ x i ě ş R y q i p dy q for all x i P M 0 . By definition of supermartingale C -monotonicity , w e get ż R ˆ R C p x, π x q µ p dx q “ 1 N N ÿ i “ 1 C p x i , p i q ď 1 N N ÿ i “ 1 C p x i , q i q “ ż R ˆ R C p x, π 1 x q µ p dx q . Prop osition 2.12. (Sufficiency) L et r ě 1 and µ, ν P P r p R q b e in c onvex de cr e asing or der, and C : R ˆ P r p R q Ñ R b e a me asur able c ost function, c ontinuous in the se c ond ar gument and such that ther e exists a finite c onstant K which satisfies, for al l p x, m q P R ˆ P r p R q , | C p x, m q | ď K ˆ 1 ` | x | r ` ż R | y | r m p dy q ˙ . L et π P Π S p µ, ν q b e sup ermartingale C -monotone w.r.t. some Γ . Then π is optimal for (WSOT). Pr o of. The argument is similar as [ 13 , Theorem 3.3], here we state it in a sligh tly simpler con text. Step 1. Supp ose µ is concen trated on a Polish subspace X of R , and the restriction C | X ˆ P r p R q is con tinuous. Then combine Lemma 2.11 with Theorem 2.3 we get π is optimal for V C S p µ, ν q . Step 2. W e lift the requirements of C b eing contin uous. Using similar approximation argumen ts with Lusin’s theorem as [ 13 , Theorem 3.3], and replace [ 13 , Lemma A.6] with [ 17 , Lemma 5.7], we can draw the conclusion. In the following, we justify the necessary side of the monotonicity principle. First, w e need an abstract version of this theorem similar as [ 10 ] with equalities in the linear constraints b eing replaced by inequalities. Then, we need to apply the abstract theorem in a suitable con text to get the monotonicity principle for (WSOT). F or completeness w e state the main results here in this App endix. Throughout this section, we denote X, Y tw o P olish spaces. 9 Let E b e a P olish space and c : E Ñ R b e Borel measurable. Given a set F of Borel functions on E , denote Π F the set of probability measures γ on E for which ş f dγ ď 0 for all f P F . In the following, w e consider the constrained optimization problem is defined by: min γ P Π F ż cdγ . F or α P M p E q , α 1 P E is a c omp etitor of α if α p E q “ α 1 p E q , and for all f P F , ş f dα “ ş f dα 1 . In addition, if α is finitely supp orted, its comp etitor should also b e so. A set Γ Ă E is called c -monotone (or finitely minimal) if each measure α , which is finite and concen trated on finitely man y p oints in Γ, is cost minimizing among its comp etitors. A measure γ is called c -monotone if it is concentrated on a c -monotone set. W e first state the adaptation of [ 10 , Theorem 1.4] into the current text, with the constrain t b eing inequalit y . Theorem 2.13. Supp ose ther e exists a function g : E Ñ r 0 , 8q such that for al l f P F , ther e exists a c onstant a f P R ` s.t. | f | ď a f g . In addition, al l functions in F ar e c ontinuous, or F is at most c ountable. Assume further γ ˚ is the optimizer of the c onstr aine d pr oblem: min γ P Π F ż cdγ “ ż cdγ ˚ P R . Then γ ˚ is c -monotone. Pr o of. As in [ 10 , Theorem 1.4], the main idea is to apply a Kellerer’s measure-theoretic duality result [ 10 , Lemma 4.1]. Define the set M similar as [ 10 , Theorem 1.4] with a sup ermartingale v ersion of c -b etter comp etitor. Also define ˆ M similar but with ÿ α 1 i f p z 1 i q ď ÿ α i f p z i q , instead of the equality . Applying the duality to M : if item (i) in the dualit y holds, then the pro of is done. Otherwise, if item (ii) holds, one try to dra w a contradiction. Indeed, by Janko v-von Neumann measurable selection theorem (see [ 15 , Prop osition 7.49] ) to the set ˆ M , one can find a mapping z ÞÑ p α 1 p z q , ¨ ¨ ¨ , α l p z q , z 1 1 p z q , ¨ ¨ ¨ , z 1 l p z q , α 1 1 p z q , ¨ ¨ ¨ , α 1 l p z qq suc h that p z , α 1 p z q , ¨ ¨ ¨ , α l p z q , z 1 1 p z q , ¨ ¨ ¨ , z 1 l p z q , α 1 1 p z q , ¨ ¨ ¨ , α 1 l p z qq P ˆ M . Argue subsequen tly similar as [ 10 , Theorem 1.4], one can then define the measures ω and ω 1 , and ω 1 is a c -b etter competitor of ω : ż f dω 1 ď ż f dω , ż cdω 1 ď ż cdω . Consequen tly we can construct another probabilit y measure γ 1 : “ γ ˚ ´ ω ` ω ˚ , whic h is also a con tradiction to the optimality of γ ˚ . 10 Remind the definition of Λ S p µ, ν q in Section 4.3. Expressed in terms of linear constraints, w e hav e P P Λ S p µ, ν q iff ż R ˆ P p R q f p x, p q P p dx, dp q ď 0 , @ f P F s , where F s : “ ! f P C b p R ˆ P 1 : D g P C ` b p P q , h P C ` b p R q , s.t. f p x, p q “ g p p q h p x q ż R p x ´ y q p p dy q ) . No w we are able to pro vide the necessity part of the monotonicit y principle. Prop osition 2.14. (Ne c essity) L et r ě 1 , and µ ď cd ν P P r p R q b e in c onvex de cr e asing or der, and C : R ˆ P r p R q Ñ R b e a me asur able c ost function, c ontinuous and c onvex in the se c ond ar gument and such that ther e exists a finite c onstant K which satisfies, for al l p x, m q P R ˆ P r p R q , | C p x, m q | ď K ˆ 1 ` | x | r ` ż R | y | r m p dy q ˙ . L et π ˚ P Π S p µ, ν q b e a sup ermartingale c oupling which minimises (WSOT), then π ˚ is a Sup ermartingale C -monotone c oupling. Pr o of. The argumen t is similar as [ 6 , Theorem 3.4], and we include it here for completeness. First we notice that (WSOT’) can b e seen as an optimal transp ort problem with addition linear constrain ts taking inequality form. By Theorem 2.13, w e ha ve if P ˚ P Λ S p µ, ν q , then P ˚ is sup ermartingale C -monotone. Recall that the sup ermartingale C -monotone is defined in Definition 2.4. Notice that any sup ermartingale coupling π P Π s p µ, ν q induces an element in Λ S p µ, ν q b y the embedding J defined b efore this prop osition; so the function v alue of (WSOT’) is smaller than (WSOT). When cost function is further conv ex as in the current con text, by Prop osition 4.3, we hav e C p x, I p Q qq ď ş P p R q C p x, p q Q p dp q . Let P P Λ S p µ, ν q , then µ p dx q I p P x q P Π S p µ, ν q . By ab o v e inequality , ż R ˆ P p R q C p x, p q P p dx, dp q ě ż R ˆ P p R q C p x, I p P x qq µ p dx q . Consequen tly the function v alues of (WSOT’) and (WSOT) are equal. As π ˚ is optimal for (WSOT), J p π ˚ q is optimal for (WSOT’); and w e deduce b y first paragraph that J p π ˚ q is sup ermartingale C-monotone. Similar reasoning as [ 6 , Remark 2.4(a)] leads to π martingale C-monotone iff J p π q martingale C-monotone. Consequently , w e ha v e π ˚ is also sup ermartingale C-monotone. 3 Pro of of Theorem 2.1 This section is dev oted to the pro of of the appro ximation theorem. First, we notice that by the same reason as [ 12 , Lemma 5.1], it is enough to prov e Theorem 2.1 in the case of r “ 1. Throughout w e fix p µ k , ν k q k ě 1 Ă P 1 ˆ P 1 suc h that W 1 p µ k , µ q Ñ 0 , W 1 p ν k , ν q Ñ 0 , µ k ď cd ν k for all k ě 1 , and w e fix a c oupling π P Π S p µ, ν q . W e construct π k P Π S p µ k , ν k q with AW 1 p π k , π q Ñ 0. 11 3.1 Outline of the pro of 3.1.1 Reduction to the strict supermartingale part (to the righ t of x ˚ ) W e recall that as in Section 4.1, w e hav e difined that D : “ P ν ´ P µ and x ˚ : “ sup t x P R : D p x q “ 0 u . W e use the irreducible decomp osition of Supermartingale coupling recalled in Lemma 4.1. It yields a decomp osition of µ “ ř n ě´ 1 µ n , ν “ ř n ě´ 1 ν n and of π “ ř n ě´ 1 π n suc h that π 0 P Π S p µ 0 , ν 0 q , π n P Π M p µ n , ν n q for all n ‰ 0 , and, for n ě 1, each pair p µ n , ν n q is an irreducible martingale component. Moreo ver µ ´ 1 “ ν ´ 1 and π ´ 1 “ µ ´ 1 p d x q δ x p d y q , whic h corresp onds to the transp ort plans on the diagonal. Prop osition 3.1. L et p µ k , ν k q k c onver ge to p µ, ν q in W 1 , wher e µ k ď cd ν k for every k . Denote by pp µ n q n ě´ 1 , p ν n q n ě´ 1 q the de c omp osition of p µ, ν q describ e d ab ove, in p articular, we have µ 0 ď cd ν 0 , µ n ď c ν n for al l n ě 1 , and µ ´ 1 “ ν ´ 1 “ η . Then ther e exist for al l k ě 1 , a de c omp osition of µ k ď cd ν k into subpr ob ability me asur es, p µ k ´ 1 , ν k ´ 1 q , p µ k 0 , ν k 0 q , p µ k n , ν k n q n ě 1 so that µ k “ µ k 0 ` µ k ´ 1 ` ÿ n ą 0 µ k n , ν k “ ν k 0 ` ν k ´ 1 ` ÿ n ą 0 ν k n , µ k 0 ď cd ν k 0 , µ k n ď cd ν k n for al l n ‰ 0 , and lim k Ñ8 µ k ´ 1 “ η , lim k Ñ8 µ k 0 “ µ 0 , lim k Ñ8 µ k n “ µ n , lim k Ñ8 ν k ´ 1 “ η , lim k Ñ8 ν k 0 “ ν 0 , lim k Ñ8 ν k n “ ν n in W 1 . In the following, we argue that it is enough to prov e Theorem 2.1 for one irreducible comp o- nen t. Indeed, applying Prop osition 3.1, w e can find subprobability measures p µ k ´ 1 , ν k ´ 1 q , p µ k 0 , ν k 0 q , p µ k n , ν k n q n ě 1 ,k ě 1 so that the desired decomp osition and order relationships are satisfied. W e treat separately the sup ermartingale comp onen t n “ 0, the martingale comp onents n ě 1 and the identical comp onen t n “ ´ 1: • w e need to get the desired approximation result for the sup ermartingale c omponent π 0 P Π S p µ 0 , ν 0 q , namely , AW 1 p π k 0 , π 0 q Ñ 0. • F or n “ ´ 1, we can argue as [ 12 , Lemma 5.2], to get the conv ergence in AW 1 p χ k , χ q Ñ 0. • F or n ě 1, notice that in general the critical p oint x ˚ ,k induced by µ k and ν k are differen t from x ˚ , hence w e cannot exp ect the appro ximating marginals p µ k n , ν k n q n ě 1 of p µ n , ν n q n ě 1 are in conv ex order. In general, they are only in con vex decreasing order. Hence the task of finding π k n P Π M p µ k n , ν k n q with AW 1 p π k n , π n q Ñ 0 will also b e solv ed by the appro ximation result. As AW 1 is additive ov er mutually singular pieces (after iden tifying the relev an t first marginals, see [12, Lemma 3.7]), we first get the appro ximation result for finitely-many comp onen ts: χ k ` π k 0 ` p ÿ n “ 1 π k n Ñ χ ` π 0 ` p ÿ n “ 1 π n in AW 1 , k Ñ `8 . 12 Com bining this with [ 12 , Lemma 3.6(b)], we finally conclude that π k “ χ k ` π k 0 ` ř p n “ 1 π k n P Π S p µ k , ν k q con verges in AW 1 to π “ χ ` π 0 ` ř p n “ 1 π n P Π S p µ, ν q . Consequen tly , from now on w e may assume that π P Π S p µ, ν q and p µ, ν q is irreducible in the sense of Lemma 4.1 . In particular, the op en set I : “ t x P R : P µ p x q ă P ν p x qu is a non-empt y op en interv al, and µ is supp orted on I : “ p l, ρ q . 3.1.2 Useful lemmas In this section, we collect some useful lemmas which will b e used in the pro of of the Theorem 2.1. Tw o preparatory regularisations. W e introduce t wo standard regularisations which preserv e the sup ermartingale prop ert y and allo w us to lo calise the coupling. F or µ, ν P P 1 , we first define tw o measures b y their put p oten tials: µ _ cd ν is the measure with put p oten tial P µ _ P ν µ ^ cd ν is the measure with put p oten tial p P µ ^ P ν q c , where h c denotes the conv ex h ull (largest con vex minoran t) of h . It is not difficult to see that ba ry p µ _ cd ν q “ bary p µ q ^ ba ry p ν q and ba ry p µ ^ cd ν q “ bary p µ q _ ba ry p ν q . Lemma 3.2 (T runcation of the k ernels) . L et π “ µ p d x q π x p d y q P Π S p µ, ν q with µ, ν P P 1 . F or R ą 0 define a kernel π R x by π R x : “ $ & % π x ^ cd ´ R ´ π x 2 R δ ´ R ` R ` π x 2 R δ R ¯ , | x | ď R, δ x , | x | ą R, and set π R : “ µ p d x q π R x p d y q with se c ond mar ginal ν R . Then π R P Π S p µ, ν R q , ν R ď c ν , and AW 1 p π R , π q Ñ 0 as R Ñ 8 . Lemma 3.3 (Affine contraction) . L et π “ µ p d x q π x p d y q P Π S p µ, ν q and let α P p 0 , 1 q . Define π α x : “ p T x,α q # π x , the push-forwar d of π x by the map T x,α q , wher e T x,α p y q : “ αy ` p 1 ´ α q x . Set π α : “ µ p d x q π α x p d y q with se c ond mar ginal ν α . Then π α P Π S p µ, ν α q and AW 1 p π α , π q ď p 1 ´ α q ´ ż R | x | µ p d x q ` ż R | y | ν p d y q ¯ . Con v ergence related to conv ex decreasing order. W e hav e the follo wing lemma. Lemma 3.4. L et p µ k q k ě 1 , p ν k q k ě 1 Ă P 1 c onver ge in W 1 to µ and ν , r esp e ctively. Then W 1 p µ k _ cd ν k , µ _ cd ν q Ñ 0 and W 1 p µ k ^ cd ν k , µ ^ cd ν q Ñ 0 . V anishing sup ermartingale defect under AW 1 . The follo wing lemma will also b e used sev eral times. Lemma 3.5. Assume π P Π S p µ, ν q and π k P Π p µ k , ν k q satisfy AW 1 p π k , π q Ñ 0 with µ k , µ P P 1 . Then ż R ´ ż R y π k x p d y q ´ x ¯ ` µ k p d x q Ý Ñ 0 . 13 3.1.3 Lo calisation and barycen tre correction Recall that I “ p l , ρ q . Fix ε P p 0 , 1 { 2 q . Cho ose a compact interv al K “ r a, b s Ă I suc h that µ p K c q ď ε . F or an y R ą 0, let p π R x q x P R b e the probabilit y kernel obtained as follo ws: if R ě | x | π R x : “ π x ^ cd ˆ R ´ π x 2 R δ ´ R ` R ` π x 2 R δ R ˙ , and π R x : “ δ x otherwise. Let π R : “ µ ˆ π R x , and π R,α x b e the image of π R x b y y ÞÑ α p y ´ x q ` x when α P p 0 , 1 q . Let π R,α x : “ µ ˆ π R,α x . Let ˜ a, ˜ b b e real num b ers suc h that ˜ a P p ℓ, a q and ˜ b P p b, ρ q . Let L b e a compact subset of I such that the in terior L ˝ of L satisfies rp´ R q _ p αℓ ` p 1 ´ α q a q , R ^ p αρ ` p 1 ´ α q b qs Ă L ˝ . W e hav e the following prop osition: Prop osition 3.6. Ther e exist R ą 0 and α P p 0 , 1 q such that AW 1 p επ ` p 1 ´ ε q π R,α , π q ă ε and α ą 2 R ´ a ´ ˜ a 2 R ´ 2˜ a _ 2 R ` b ` ˜ b 2 R ` 2 ˜ b . µ | K ˆ π R,α x is c onc entr ate d on K ˆ L ˝ . Denote the se c ond mar ginal of π R,α by ν R,α . ν R,α ˆˆ ℓ, a ` ˜ a 2 ˙˙ ą 0 . T o summarise, we hav e constructed a sup ermartingale coupling π R,α P Π S p µ, ν R,α q close to π under AW 1 , whose restriction π R,α | K ˆ R is compactly supp orted on K ˆ L and concen trated on K ˆ L ˝ . Moreo ver, the second marginal distribution ν R,α is dominated by ν in ď c and assigns p ositiv e mass on the left side of K . Preparing target measures for the appro ximating sequence. W e will work with an in termediate target measure ν R,α,k satisfying µ k ď cd ν R,α,k and ν R,α,k ď c ν k , Let ∆ k : “ bary p ν k q ´ ba ry p ν R,α q and denote by T ∆ k the translation x ÞÑ x ` ∆ k . Define ν R,α,k : “ ν k ^ c ` µ k _ cd T ∆ k # ν R,α ˘ . (10) Then Lemma 3.4 implies W 1 p ν R,α,k , ν R,α q Ñ 0 and, b y construction, µ k ď cd ν R,α,k ď c ν k for all k . Lo calising the coupling. Next, we hav e to adjust the barycen tres of its disin tegrations, π R,α,k x to obtain sup ermartingale k ernels and thereby sup ermartingale couplings. Due to the inner regularity of ν R,α , we find compact set L ´ Ă ` ℓ, a ` ˜ a 2 ˘ with p ositiv e measure under ν R,α . Let ˜ ℓ, ˜ ρ P I , b e suc h that ˜ ℓ ă inf p L Y L ´ q and ˜ ρ ą sup p L Y L ` q . Then define ˜ L ´ : “ ´ ˜ ℓ, a ` ˜ a 2 ¯ and ˜ K : “ ´ 3 a ` ˜ a 4 , 3 b ` ˜ b 4 ¯ so that ˜ L ´ and ˜ K are b ounded and open interv als co vering respectively L ´ and K , and that the distance e b et w een ˜ L ´ Y ˜ L ` and ˜ K is p ositiv e: Set J : “ r ˜ ℓ, ˜ ρ s . 14 The resp ectiv e restriction of ν R,α,k to ˜ L ´ and ˜ L ` are denoted b y ν k ´ and ν k ` . Since ˜ L ´ and ˜ L ` are op en, P ortmanteau’s theorem ensures that for k sufficien tly large ν k ´ and ν k ` eac h ha ve more total mass than some constan t δ ą 0. Applying Lemma 4.2 (ii) with A : “ K , B : “ ˜ K , Y : “ R , C : “ L ˝ , there are ˆ µ k ď µ k , ˆ ν k ď ν R,α,k , ˆ π k : “ ˆ µ k ˆ ˆ π k x P P p ˆ µ k , ˆ ν k q concen trated on ˜ K ˆ L ˝ , and ε k ě 0 suc h that AW 1 ` ˆ π k , p 1 ´ ε k q π R,α | K ˆ R ˘ Ñ 0 , as k Ñ 8 . (11) Correcting the sup ermartingale constrain t. The k ernels ˆ π k x need not satisfy ş y ˆ π k x p d y q ď x . W e correct them by mixing in a small amoun t of mass from the left of K . Pic k an op en in terv al L ´ Ă I with sup L ´ ă a and ν R,α p L ´ q ą 0 (irreducibility ensures ν and hence ν R,α c harge every neighbourho o d of the left endp oin t of I ). Let ν R,α,k ´ : “ ν R,α,k ˇ ˇ L ´ . By P ortmanteau, ν R,α,k ´ p L ´ q ě δ ą 0 for all k large enough. F or such k and for ˆ µ k -a.e. x , define c k p x q : “ ` ş y ˆ π k x p d y q ´ x ˘ ` ş p x ´ y q ν R,α,k ´ p d y q P r 0 , 8q , d k p x q : “ 1 ` c k p x q ν R,α,k ´ p R q , and set the corrected kernel ˜ π k x : “ ˆ π k x ` c k p x q ν R,α,k ´ d k p x q P P 1 , ˜ π k : “ ˆ µ k p d x q ˜ π k x p d y q . (12) W e hav e the following prop osition. Prop osition 3.7. ˜ π k is a (sub-pr ob ability) sup ermartingale c oupling b etwe en ˆ µ k and its se c ond mar ginal ˜ ν k . In addition, AW 1 ´ ˜ π k , p 1 ´ ε k q π R,α ˇ ˇ K ˆ R ¯ Ý Ñ 0 . (13) Mor e over, for k lar ge enough, the se c ond mar ginal ˜ ν k satisfies the domination p 1 ´ 2 ε q ˜ ν k ď p 1 ´ ε q ν R,α,k . (14) 3.1.4 Completion and gluing W e now complement the sup ermartingale coupling p 1 ´ 2 ε q ˜ π k (whic h is a sub-coupling b etw een p 1 ´ 2 ε q ˆ µ k and p 1 ´ 2 ε q ˜ ν k ) to a sup ermartingale coupling with marginals µ k and εν k `p 1 ´ ε q ν R,α,k for k sufficiently large. Let µ k rem : “ µ k ´ p 1 ´ 2 ε q ˆ µ k , ν k rem : “ εν k ` p 1 ´ ε q ν R,α,k ´ p 1 ´ 2 ε q ˜ ν k . By (14) we ha ve ν k rem ě 0. In the follo wing, we shall prov e that: µ k rem ď cd ν k rem . (15) W e first notice that it is easy to prov e that the result holds on J . 15 Prop osition 3.8. It is enough to pr ove (15) on J c . It remains to deal with the complement of J , and w e prov e equiv alen tly: P εν k `p 1 ´ ε q ν R,α,k ´ P µ k ě P p 1 ´ 2 ε q ˜ ν k ´ P p 1 ´ 2 ε q ˆ µ k . Indeed, in the following we will pro ve a slightly stronger result: Prop osition 3.9. We have that P ν R,α,k ´ P µ k ě P p 1 ´ 2 ε q ˜ ν k ´ P p 1 ´ 2 ε q ˆ µ k on J c . Remark 3.10. It is w orth noting that, the similar inequality of [ 12 ] holds in the martingale con text, i.e. the l.h.s. ab o ve is p ositiv e, and the r.h.s. is n ull. F or the sup ermartingale case, b oth sides are p ositive, and one needs to carry out tailored analysis using the definitions of all quan tities inv olv ed. It turns out that the ε ’s in the inequalit y are crucial for the proof as it giv es ro oms for some relaxation. By Strassen’s theorem for the supermartingale order, (15) implies that there exists η k P Π S p µ k rem , ν k rem q . Define the glued coupling ¯ π k : “ p 1 ´ 2 ε q ˜ π k ` η k P Π S ´ µ k , εν k ` p 1 ´ ε q ν R,α,k ¯ . Com bining (13) with the choice of K, R, α and the fact that µ p K c q ď ε , we obtain lim sup k Ñ8 AW 1 p ¯ π k , π q ď C ε, (16) for a finite constant C dep ending only on the first moments of µ and ν (one ma y take C as in the gluing estimate of [12, Lemma 3.6]). 3.1.5 Final adjustmen t of the second marginal The second marginal of ¯ π k equals εν k ` p 1 ´ ε q ν R,α,k , whic h satisfies εν k ` p 1 ´ ε q ν R,α,k ď c ν k since ν R,α,k ď c ν k . Hence there exists a martingale coupling M k P Π M p εν k ` p 1 ´ ε q ν R,α,k , ν k q . Moreov er, by a quantitativ e v ersion of Strassen’s theorem on the line (see, e.g., [12, Theorem 2.12]), we can choose M k suc h that ż R 2 | z ´ y | M k p d z , d y q ď 2 W 1 ´ εν k ` p 1 ´ ε q ν R,α,k , ν k ¯ . The righ t-hand side tends to 2 ε W 1 p ν k , ν k q ` 2 p 1 ´ ε q W 1 p ν R,α,k , ν k q and can be made arbitrarily small b y first taking k Ñ 8 and then ε Ó 0 (since ν R,α,k Ñ ν R,α and ν R,α Ñ ν ). Finally define π k P Π S p µ k , ν k q b y comp osing ¯ π k with M k on the second co ordinate: π k p d x, d y q : “ ż R ¯ π k p d x, d z q M k z p d y q . This preserves the sup ermartingale constrain t (comp osition of a sup ermartingale step with a martingale step remains sup ermartingale), and w e hav e the estimate AW 1 p π k , ¯ π k q ď ż R 2 | z ´ y | M k p d z , d y q . Com bining with (16) and sending first k Ñ 8 and then ε Ó 0 yields AW 1 p π k , π q Ñ 0. This concludes the pro of of Theorem 2.1. 16 3.2 Pro ofs of the relev an t results 3.2.1 Pro of of the results in Section 3.1.1 Pr o of of Pr op osition 3.1 F or all k ě 1, pic k a coupling π k P Π S p µ k , ν k q , let x ˚ ,k : “ sup t x : P µ k “ P ν k u , and l n , r n b e the other b oundaries of t x : p µ n ă p ν n u (excluding x ˚ ). Although in general, x ˚ ,k ‰ x ˚ . W e can use a construction similar to [ 12 , Proposition 2.5]. F or the first marginal in the appro ximating sequence, one uses the limiting coupling’s irreducible comp onen ts; while for the second marginal, one tak es adv an tage of the k ernel of the appro ximating coupling. F or n ě 1, we define: µ k n : “ ż F µ p r n ´q u “ F µ p l n q δ F ´ 1 µ k p u q p dx q du and ν k n : “ ż F µ p r n ´q u “ F µ . p l n q π k F ´ 1 µ k p u q p dy q du F or n “ ´ 1, w e replace ab o v e l n , r n with resp ectiv ely x ˚ , 8 to define µ k ´ 1 and ν k ´ 1 . Finally we can set J 1 : “ r 0 , 1 szp F µ p x ˚ q , 1 q and J : “ J 1 z ď n P N p F µ p l n q , F µ p r n ´qq , and define η k p dx q : “ ż u P J δ F ´ 1 µ k p u q p dx q du and v k p dx q : “ ż u P J π k F ´ 1 µ k p u q p dy q du. By the same argumen ts as [ 12 , Prop osition 2.5], w e can prov e these measures satisfied the desired decomp osition and con vex decreasing order prop erties. 3.2.2 Pro of of the results in Section 3.1.2 Pr o of of L emma 3.2. F or eac h x , the measure π R x is supp orted on r´ R, R s and satisfies π R x ď cd π x (b y construction of ^ cd ), hence ş y π R x p d y q ď ş y π x p d y q ď x ; therefore π R is a sup ermartingale coupling. Moreo ver, for each x w e hav e W p π R x , π x q Ñ 0 as R Ñ 8 and the b ound W p π R x , π x q ď 2 ş | y | π x p d y q . In tegrating against µ and using dominated conv ergence yields ş W p π R x , π x q µ p d x q Ñ 0 and thus A W 1 p π R , π q Ñ 0. Finally , ν R “ ş π R x µ p d x q and ν R ď c ν follo ws from π R x ď c π x for all x . Pr o of of L emma 3.3. W e ha ve ş y π α x p d y q “ α ş y π x p d y q ` p 1 ´ α q x ď x , hence π α is a sup ermartingale coupling. F or the distance b ound, use the coupling p x, y q ÞÑ p x, T x,α p y qq b et w een π and π α and note that | y ´ T x,α p y q| “ p 1 ´ α q| y ´ x | ; taking exp ectations giv es the claimed b ound. Pr o of of L emma 3.4. Observ e that, for eac h y P R , all three functions x ÞÑ | y ´ x | , x ÞÑ p y ´ x q ` and x ÞÑ x are 1-Lipsc hitz. Hence, since p µ k q k ě 1 and p ν k q k ě 1 con verge in W 1 to µ and ν , resp ectiv ely , we ha ve that, for eac h y P R , U µ k p y q , U ν k p y q , P µ k p y q , P ν k p y q , µ k , ν k con verges to U µ p y q , U ν p y q , P µ p y q , P ν p y q , µ, ν , resp ectiv ely . W e first deal with the con vergence of µ k _ cd ν k to µ _ cd ν . Since, for each y P R , lim k Ñ8 P µ k _ cd ν k p y q “ lim k Ñ8 p P µ k p y q _ P ν k p y qq “ P µ p y q _ P ν p y q “ P µ _ cd ν p y q , 17 lim k Ñ8 µ k _ cd ν k “ lim k Ñ8 p µ k ^ ν k q “ µ ^ ν “ µ _ cd ν , w e hav e that lim k Ñ8 U µ k _ cd ν k p y q “ lim k Ñ8 p 2 P µ k _ cd ν k p y q ` µ k _ cd ν k ´ y q “ 2 P µ _ cd ν p y q ` µ _ cd ν ´ y “ U µ _ cd ν p y q . This p oin t wise conv ergence of p otentials do es in fact imply the conv ergence of µ k _ cd ν k to µ _ cd ν in W 1 . Indeed, we ha ve the conv ergence of the first moments: lim k Ñ8 ż R | x | p µ k _ cd ν k qp dx q “ lim k Ñ8 U µ k _ cd ν k p 0 q “ U µ _ cd ν p 0 q “ ż R | x | p µ _ cd ν qp dx q . On the other hand, b y Chacon [ 18 , Lemma 2.6], w e also hav e that µ k _ cd ν k con verges to µ _ cd ν w eakly . Since the conv ergence of W 1 is equiv alent to the w eak con vergence together with the conv ergence of the first momen ts, the claim follows. In a similar w ay we can establish the con vergence of µ k ^ ν k to µ ^ ν in W 1 . Again, the goal is to show that the p oten tials U µ k ^ ν k con verge to U µ ^ ν p oin t wise. Note that lim k Ñ8 µ k ^ ν k “ lim k Ñ8 p µ k _ ν k q “ µ _ ν “ µ ^ ν . Hence, since for each y P R , U µ k ^ ν k p y q “ 2 P µ k ^ ν k p y q ` µ k ^ ν k ´ y , it is enough to show that lim k Ñ8 P µ k ^ ν k p y q “ P µ ^ ν p y q , for each y P R . First note that, for η P t µ, ν u , w e hav e that P η k con verges to P η uniformly on R . Indeed, for eac h y P R , | P η k p y q ´ P η p y q | “ 1 2 | U η k p y q ` y ´ η k ´ U η p y q ´ y ` η | ď 1 2 | U η k p y q ´ U η p y q | ` 1 2 | η k ´ η | ď W 1 p η k , η q , and therefore sup y P R | P η k p y q ´ P η p y q | ď W 1 p η k , η q . Then it follo ws that P µ k ^ P ν k also con verges uniformly on R to P µ ^ P ν . Let ϵ ą 0 and k 0 P N b e such that, for all k ě k 0 , sup y P R | p P µ k ^ P ν k qp y q ´ p P µ ^ P ν qp y q | ď ϵ . Then w e hav e b oth p P µ ^ P ν q c ´ ϵ ď p P µ ^ P ν q ´ ϵ ď P µ k ^ P ν k , p P µ k ^ P ν k q c ´ ϵ ď p P µ k ^ P ν k q ´ ϵ ď P µ ^ P ν , whic h then imply p P µ ^ P ν q c ´ ϵ ď p P µ k ^ P ν k q c , p P µ k ^ P ν k q c ď p P µ ^ P ν q c ` ϵ, resp ectiv ely (where we used that the conv ex h ull is the largest conv ex minoran t of a function). It follo ws that p P µ ^ P ν q c ´ ϵ ď p P µ k ^ P ν k q c ď p P µ ^ P ν q c ` ϵ. 18 By sending k Ñ 8 , and using that ϵ ą 0 w as arbitrary , w e conclude that lim k Ñ8 p P µ k ^ P ν k q c “ p P µ ^ P ν q c . Pr o of of L emma 3.5. Let χ k P Π p µ k , µ q b e an optimal coupling for A W 1 p π k , π q . Using the fact that ş y π x 1 p d y q ď x 1 for µ -a.e. x 1 and | ş y π k x ´ ş y π x 1 | ď W p π k x , π x 1 q , w e hav e ż R p ż R y π k x p dy q ´ x q ` µ k p dx q “ ż R p ż R y π k x p dy q ´ x q ` χ k p dx, dx 1 q ď ż R „ p ż R y π k x p dy q ´ x 1 q ` ` p x 1 ´ x q ` ȷ χ k p dx, dx 1 q ď ż R „ p ż R y π k x p dy q ´ ż R y π x 1 p dy qq ` ` p x 1 ´ x q ` ȷ χ k p dx, dx 1 q ď ż R ” | x 1 ´ x | ` W p π k x , π x 1 q ı χ k p dx, dx 1 q ď AW 1 p π , π k q . 3.2.3 Pro of of the results in Section 3.1.3 Pro of of Prop osition 3.6 By construction, π R x ď cd π x for all x P R , and π R x is concen trated on r´ R, R s for R ě | x | . In particular, W 1 p π R x , π x q Ñ 0 as R Ñ 8 , and W 1 p π R x , π x q ď 2 ż R | y | π x p d y q . Let π R : “ µ ˆ π R x , then dominated conv ergence yields AW 1 p π R , π q ď ż R W 1 p π R x , π x q µ x p d x q Ñ 0 , as R Ñ 8 . Denote b y ν R the second marginal of π R . Consequently , ν R con verges to ν under W 1 and ν R ď c ν (or ν R ď cd ν ) for all R ą 0. Let ˜ a, ˜ b b e real num bers such that ˜ a P p ℓ, a q and ˜ b P p b, ρ q . F or any z P R such that ν pp´8 , z sq “ 0, one has 0 ď ż R p z ´ x q ` µ p d x q ď ż R p z ´ x q ` ν p d x q “ 0 , and th us z R I as p µ p z q “ p ν p z q . Since p µ, ν q is irreducible on I , w e deduce that ν m ust assign p ositiv e mass to an y neighbourho o d of ℓ . F rom no w on, we use the notational con ven tion that for all c P R Y t´8 , 8u , r´8 , c q : “ t x P R : x ă c u , p c, ´8s : “ t x P R : x ą c u , r´8 , 8s : “ R . Thanks to the ab o ve conv en tion, one has I “ r ℓ, ρ s Ă R . 19 Then r ℓ, ˜ a q is relatively op en on I with ν R p I q “ 1 “ ν p I q , so P ortmanteau’s theorem yields lim inf R Ñ8 ν R ` r ℓ, ˜ a q ˘ ě ν ` r ℓ, ˜ a q ˘ ą 0 . Hence, w e may pic k R ą 0 large enough such that R ą | a | _ | b | , ż R W 1 p π R x , π x q µ x p d x q ă ε, ν R ` r ℓ, ˜ a q ˘ ą 0 . Let π R,α x b e the image of π R x b y y ÞÑ α p y ´ x q ` x when α P p 0 , 1 q . Then π R,α : “ µ ˆ π R,α x satisfies b y Lemma 4.2 AW 1 p επ ` p 1 ´ ε q π R,α , π q ď p 1 ´ α q ˆ ż R | x | µ p d x q ` ż R | y | ν p d y q ˙ ` ż R W 1 p π R x , π x q µ p d x q , where the right-hand side conv erges to ż R W 1 p π R x , π x q µ p d x q ă ε as α Ñ 1. Note 2 R ´ a ´ ˜ a 2 R ´ 2˜ a , 2 R ` b ` ˜ b 2 R ` 2 ˜ b P p 0 , 1 q , so w e can choose α P p 0 , 1 q suc h that AW 1 p επ ` p 1 ´ ε q π R,α , π q ă ε and α ą 2 R ´ a ´ ˜ a 2 R ´ 2˜ a _ 2 R ` b ` ˜ b 2 R ` 2 ˜ b . Let L b e a compact subset of I such that the in terior L ˝ of L satisfies rp´ R q _ p αℓ ` p 1 ´ α q a q , R ^ p αρ ` p 1 ´ α q b qs Ă L ˝ . Because R ě p´ a q _ b and thereb y r a, b s “ K Ă r´ R, R s , w e ha v e that µ | K ˆ π R x is concen trated on K ˆ pr´ R, R sX I q . F urthermore, for any p x, y q P K ˆ pr´ R, R sX I q , w e find αy ` p 1 ´ α q x P L ˝ . Hence, µ | K ˆ π R,α x is concen trated on K ˆ L ˝ . Denote the second marginal of π R,α b y ν R,α . Since p x, y q P p ℓ, R q ˆ r ℓ, ˜ a s ù ñ ℓ ă p 1 ´ α q x ` αy ă R ´ α p R ´ ˜ a q ď a ` ˜ a 2 , w e hav e that ν R,α ˆˆ ℓ, a ` ˜ a 2 ˙˙ ě ż p ℓ,R q π R x pp´8 , ˜ a qq µ p d x q “ ż R π R x pp´8 , ˜ a qq µ p d x q “ ν R pp´8 , ˜ a qq “ ν R pr ℓ, ˜ a qq ą 0 . T o summarise , w e hav e constructed a sup ermartingale coupling π R,α P Π S p µ, ν R,α q close to π under AW 1 , whose restriction π R,α | K ˆ R is compactly supp orted on K ˆ L and concen trated on K ˆ L ˝ . Moreo ver, the second marginal distribution ν R,α is dominated by ν in ď c and assigns p ositiv e mass on the left side of K . 20 Pro of for Prop osition 3.7. In order to mak e ˜ π k x b e a probabilit y measure with mean less than x , we should imp ose 1 ` c k p x q ν R,α,k ´ p R q “ d k p x q ż R y ˆ π k x p d y q ` c k p x q ż R y ν R,α,k ´ p d y q ď xd k p x q , whic h leads to c k p x q „ ν R,α,k ´ p R q x ´ ż R y ν R,α,k ´ p d y q ȷ ě ż R y ˆ π k x p d y q ´ x. Note that ν R,α,k ´ p R q x ´ ż R y ν R,α,k ´ p d y q ě ν R,α,k ´ p R q e ě δ e. W e may set th us c k p x q : “ p ş R y ˆ π k x p d y q ´ x q ` ş R p x ´ y q ν R,α,k ´ p d y q P « 0 , p ş R y ˆ π k x p d y q ´ x q ` ν R,α,k ´ p R q e ff d k p x q : “ 1 ` c k p x q ν R,α,k ´ p R q P « 1 , 1 ` p ş R y ˆ π k x p d y q ´ x q ` ν R,α,k ´ p R q e ff . Remem b er that L Y ˜ L ´ Y ˜ L ` Ă r ˜ ℓ, ˜ ρ s Ă I . Then we obtain for µ k p d x q´ almost every x the estimate W 1 p ˜ π k x , ˆ π k x q “ W 1 ˜ ˆ π k x ` c k p x q ν R,α,k ´ d k p x q , ˆ π k x ¸ ď W 1 ˆ c k p x q d k p x q ν R,α,k ´ , c k p x q d k p x q ˆ π k x ˙ ď p ş R y ˆ π k x p d y q ´ x q ` e p ˜ ℓ ´ ˜ ρ q . Hence, the adapted W asserstein distance b et ween ˆ π k and ˜ π k “ ˆ µ k ˆ ˜ π k x satisfies AW 1 p ˜ π k , ˆ π k q ď ż R W 1 p ˜ π k x , ˆ π k x q ˆ µ k p d x q ď p ˜ ℓ ´ ˜ ρ q e ż R ˆ ż R y ˆ π k x p d y q ´ x ˙ ` ˆ µ k p d x q ď p ˜ ℓ ´ ˜ ρ q e AW 1 ` ˆ π k , p 1 ´ ε k q π R,α | K ˆ R ˘ , where the last inequality follows from Lemma 3.5. The triangle inequality then yields lim k Ñ8 AW 1 ` ˜ π k , p 1 ´ ε k q π R,α | K ˆ R ˘ “ 0 . Next w e b ound the total mass which w e require to fix the barycentres. W e find that ż R c k p x q d k p x q ˆ µ k p d x q ď 1 δ e ż R ˆ ż R y ˆ π k x p d y q ´ x ˙ ` ˆ µ k p d x q Ñ 0 , as k Ñ 8 . Consequen tly , when ˜ ν k denotes the second marginal of ˜ π k , w e ha ve for k sufficien tly large that p 1 ´ 2 ε q ˜ ν k “ 1 ´ 2 ε d k p x q ˆ ν k ` p 1 ´ 2 ε q ν R,α,k ´ ż R c k p x q d k p x q ˆ µ k p d x q ď p 1 ´ 2 ε q ˆ ν k ` p 1 ´ 2 ε q ν R,α,k ε ď p 1 ´ 2 ε q ν R,α,k ` εν R,α,k “ p 1 ´ ε q ν R,α,k . 21 3.2.4 Pro of of the results in Section 3.1.4 Pr o of of Pr op osition 3.8. Recall that ˜ π k P Π S p ˆ µ k , ˜ ν k q and π R,α | K ˆ R P Π S p µ | K , ˘ ν R,α q , where ˘ ν R,α is the second marginal distribution of π R,α | K ˆ R , are concentrated on the compact cub e J ˆ J and from (13) AW 1 p ˜ π k , p 1 ´ ε k q π R,α | K ˆ R q Ñ 0 , as k Ñ 8 . F urthermore, since p 1 ´ ε q π R,α ´ p 1 ´ 2 ε q π R,α | K ˆ R is a supermartingale coupling with marginals p 1 ´ ε q µ ´ p 1 ´ 2 ε q µ | K and p 1 ´ ε q ν R,α ´ p 1 ´ 2 ε q ˘ ν R,α , w e deduce by the irreducibility of pair p µ, ν q on I the irreducibility of the sub-probabilit y measures εµ ` p 1 ´ ε q µ ´ p 1 ´ 2 ε q µ | K and εν ` p 1 ´ ε q ν R,α ´ p 1 ´ 2 ε q ˘ ν R,α , whose put functions satisfy 0 ď P µ ´ P p 1 ´ 2 ε q µ | K ă P εν `p 1 ´ ε q ν R,α ´ P p 1 ´ 2 ε q ˘ ν R,α on I . Since those potential functions are contin uous, there exists τ ą 0 suc h that they ha ve distance greater than τ on J . By the uniform conv ergence of p oten tial functions, for k sufficien tly large w e hav e 0 ď P µ k ´ P p 1 ´ 2 ε q ˆ µ k ` τ 2 ď P εν k `p 1 ´ ε q ν R,α,k ´ P p 1 ´ 2 ε q ˜ ν k on J. Pr o of of Pr op osition 3.9. First, the result on the left hand side of J is clear. On the righ t hand side of J , as b oth p oten tial functions p p 1 ´ 2 ε q ˜ ν k and p p 1 ´ 2 ε q ˆ µ k ha ve touc hed their asymptotic lines, we ha ve P p 1 ´ 2 ε q ˜ ν k p x q ´ P p 1 ´ 2 ε q ˆ µ k p x q “ p 1 ´ 2 ε q ˆ ż R p x ´ y q ˜ ν k p d y q ´ ż R p x ´ y q ˆ µ k p d y q ˙ “ p 1 ´ 2 ε qr ˆ µ k ´ ˜ ν k s . In order to pro ceed the pro of, we remind the definitions of some auxiliary quan tities following notations Lemma 4.2 of [12] (with slight minor c hanges): ˆ µ k “ : 1 ´ ε 1 k 1 ´ ε k ˜ µ k ,  ν k : “ ż R π R,α,k x ˜ µ k p d x q , and w e recall ˜ µ k ď µ k | ˜ K , ˆ ν k “  ν k | L , where on the abov e, ε k and ε 1 k are t wo small terms satisfying that ε k k Ñ8 Ñ 0, ε 1 k k Ñ8 Ñ 0, 1 ´ ε 1 k 1 ´ ε k ď 1 for all k ě 1, and π R,α | K ˆ R is supp orted on K ˆ L ˝ . W e also note that ν R,α,k “ ż R π R,α,k x µ k p d x q . 22 In addition, ˜ ν k “ 1 d k p x q ˆ ν k ` ν R,α,k ´ ż R c k p x q d k p x q ˆ µ k p d x q . W e shall decomp ose the pro of in four steps, and prov e that when k is large enough: P ν R,α,k ´ P µ k p i q ě p 1 ´ ε 2 qp µ k ´ ν R,α,k q p ii q ě p 1 ´ ε 2 qp ˜ µ k ´  ν k q ´ ε 2 p ˆ µ k ´ ˜ ν k q p iii q ě p 1 ´ ε qp ˆ µ k ´ ˆ ν k q ´ ε p ˆ µ k ´ ˜ ν k q p iv q ě p 1 ´ 2 ε qp ˆ µ k ´ ˜ ν k q . In the follo wing, w e shall address each inequality (i)-(iv) separately . W e first introduce tw o lemmas. The first Lemma is in the same spirit of Remark 2.2 in [12]. Throughout this pro of, w e will rep eatedly use the following fact: Lemma 3.11. L et ϵ ą 0 , p a k q k ě 1 and p b k q k ě 1 b e two se quenc es of r e al numb ers such that a k Ñ 0 , b k Ñ b ą 0 , then we have for k lar ge enough, a k ď εb k . Pr o of. The result is clear, and w e omit the details. First, we consider step (ii). Indeed, ν R,α,k and  ν k are related to µ k and ˜ µ k b y conv olution with the same kernel: ν R,α,k “ ż π R,α,k x µ k p dx q ,  ν k “ ż π R,α,k x ˜ µ k p d x q . In addition, π R,α,k x con verges in W 1 to π R,α x , whic h is a s upermartingale measure. ż R p ż R y π R,α,k x p dy q ´ x q ` µ k p dx q Ñ 0 . As ˜ µ k ď µ k | K , it follows that (denoting δ µ k : “ µ k ´ ˜ µ k ) ´ µ k ´ ν R,α,k ¯ ´ ´ ˜ µ k ´  ν k ¯ “ δ µ k ´ ż π R,α,k x δ µ k p dx q “ ż R p x ´ ż R y π R,α,k x p dx qq δ µ k p dx q ď ż R p ż R y π R,α,k x p dx q ´ x q ` δ µ k p dx q Ñ 0 . Ab o v e in the last line we hav e applied Lemma 3.5 and the fact that ´ a “ a ´ ´ a ` ď a ´ , for an y a P R . As ˆ µ k Ñ µ | K , ˜ ν k Ñ q ν R,α , it follows that p ˆ µ k ´ ˜ ν k q Ñ p µ | K ´ q ν R,α q ą 0 . It follo ws from Lemma 3.11 with a k “ ´ µ k ´ ν R,α,k ¯ ´ ´ ˜ µ k ´  ν k ¯ , and b k “ ˆ µ k ´ ˜ ν k that inequalit y (ii) is v alid. 23 Second, to v erify (iv), w e calculate the difference ˜ ν k ´ ˆ ν k . Using the definition of ˜ ν k in the previous page, we get ˜ ν k ´ ˆ ν k “ p 1 d k p x q ´ 1 q ˆ ν k ` ν R,α,k ´ ż R c k p x q d k p x q ˆ µ k p d x q : “ β 1 ` β 2 . W e analyse the t wo terms separately . β 2 ď | ˜ l | ¨ ż R c k p x q d k p x q ˆ µ k p d x q Ñ 0 , when k Ñ 8 . On the abov e w e ha ve used the fact that | ş R xν R,α,k ´ p d x q| ď | ˜ l | , where ˜ l is the left endp oin t for J . And the con vergence is v alid due to the second from last inequality in step 2. In addition, for β 1 , w e hav e |p 1 ´ 1 { d k p x qq| ˆ ν k “ | c k p x q ν R,α,k ´ p R q d k p x q ˆ ν k | ď | ˆ ν k ν k ´ p R q e | ¨ ˆ ż R y ˆ π k x p dy q ´ x ˙ ` , whic h conv erges ˆ µ k p dx q almost ev erywhere to 0(the reason comes also from second from last inequalit y in step 2) when k Ñ 8 . Consequen tly β 1 Ñ 0. Hence the difference of replacing ˜ ν k with ˆ ν k can b e absorbed into the ε terms. It follo ws from Lemma 3.11 with a k “ ´ ˆ µ k ´ ˆ ν k ¯ ´ ´ ˆ µ k ´ ˜ ν k ¯ , and b k “ ˆ µ k ´ ˜ ν k that inequalit y (iv) is v alid. Third, to verify (iii), we recall that ˆ µ k “ 1 ´ ε 1 k 1 ´ ε k ˜ µ k “ β k ˜ µ k ď ˜ µ k . In particular, on ab o ve β k : “ 1 ´ ε 1 k 1 ´ ε k “ ˆ µ k p L q ˜ µ k p R q (see Lemma 4.2). Now in order that p 1 ´ ε qp ˜ µ k ´  ν k qěp 1 ´ 3 2 ε qp ˆ µ k ´ ˆ ν k q , it is enough that ˆ ν k ě β k  ν k , whic h is equiv alent to ˆ ν k  ν k ě ˆ µ k p L q ˜ µ k p R q “ ˆ µ k ˜ µ k “  ν k p L q  ν k p R q , whic h is further equiv alent to ba ry p ˆ ν k “  ν k | L q ě bary p  ν k q . W e recall that ˜ µ k Ñ µ | K , and  ν k Ñ µ | K b π R,α x , whic h is supp orted on L . In addition,  ν k “ ˜ µ k b π R,α,k x . W e get that when k Ñ 8 , ba ry p ˆ ν k “  ν k | L q ´ ba ry p  ν k q Ý Ñ 0 . Using similar p oin ts when verifying p oint (iv), we confirm (iii). 24 F or p oin t (i), it comes to the choice of the righ t endp oin t of J , whic h is ˜ ρ . Indeed, when x Ñ ρ , we hav e b oth p ν R,α,k and p µ k touc h the asymptotic line and hence lim x Ñ ρ p p ν R,α,k ´ p µ k qp x q ´ p µ k ´ ν R,α,k q “ 0. W e can choose ˜ ρ close enough to ρ suc h that @ x P r ˜ ρ, ρ s , we hav e p p ν R,α,k ´ p µ k q ´ p µ k ´ ν R,α,k qp x q ď ε 2 p µ k ´ ν R,α,k q . Here w e used the fact that if the functions f k and f are con tinuous, f k Ñ f uniformly , and lim x Ñ ρ f p x q “ 0. Then D ˜ ρ , such that for all k ě k 1 , f k p x q ď ε 2 , for x P r ˜ ρ, ρ s . 4 App endix In this App endix, w e collect some needed results. 4.1 Irreducible decomp osition for sup ermartingale couplings Let µ, ν P P 1 with µ ď cd ν and set D : “ P ν ´ P µ . Define x ˚ : “ sup t x P R : D p x q “ 0 u . Roughly sp eaking, p oin ts where D v anishes act as barriers: for π P Π S p µ, ν q , no mass can cross such a p oin t on the “martingale side”; see [ 32 ]. This yields an irreducible decomp osition in to comp onents where the problem is gen uinely sup ermartingale on the right and martingale on the left. Lemma 4.1 (Nutz–Stebegg [ 32 ],Prop osition 3.4) . L et µ, ν P P 1 with µ ď cd ν . L et I 0 : “ p x ˚ , `8q , let p I k q k ě 1 b e the op en c onne cte d c omp onents of t D ą 0 u X p´8 , x ˚ q , and set I ´ 1 : “ R z Ť k ě 0 I k . L et µ k : “ µ | I k for k ě ´ 1 , so that µ “ ř k ě´ 1 µ k . Then ther e exists a unique de c omp osition ν “ ř k ě´ 1 ν k such that µ ´ 1 “ ν ´ 1 , µ 0 ď cd ν 0 and µ k ď c ν k for al l k ě 1 . F urthermor e, any π P Π S p µ, ν q admits a unique de c omp osition π “ ř k ě´ 1 π k such that π 0 P Π S p µ 0 , ν 0 q and π k P Π M p µ k , ν k q for al l k ‰ 0 . 4.2 A lo calisation lemma in adapted W asserstein top ology The following result (from [ 12 ]) allows one to lo calise adapted W asserstein conv ergence on sets of p ositiv e mass. It is a k ey technical input for our appro ximation argument. Lemma 4.2 ([ 12 ], Lemma 3.4) . L et r ě 1 . L et µ, µ k P M r p X q and ν, ν k P M r p Y q have e qual masses, and assume that π k P Π p µ k , ν k q AW 1 Ý Ý Ý Ñ π P Π p µ, ν q . L et A Ă X b e me asur able and let B Ą A b e op en. (i) Let γ k P Π p µ k , µ q b e an optimizer in (7) for A W 1 p π k , π q and set ˜ µ k : “ γ k p¨ ˆ A q , ε k : “ 1 ´ ˜ µ k p X q µ p A q . 25 Then ˜ µ k ď µ k | B and ε k ě 0. Moreo v er, defining ˜ π k : “ ˜ µ k ˆ π k x , one has AW 1 ` ˜ π k , p 1 ´ ε k q π | A ˆ Y ˘ ` ε k Ý Ñ 0 . (ii) Supp ose that ν is concentrated on some C Ă Y . Let ˜ ν k and ˜ ν b e the second marginals of ˜ π k and π | A ˆ Y , resp ectiv ely , and define ˆ µ k “ ˜ ν k p C q ˜ µ k p X q ˜ µ k , ε 1 k : “ 1 ´ ˜ ν k p C q µ p A q . Then there exist ˆ ν k ď ν k and ˆ π k “ ˆ µ k ˆ ˆ π k x P Π p ˆ µ k , ˆ ν k q concen trated on B ˆ C suc h that AW r 1 ` ˆ π k , p 1 ´ ε 1 k q π | A ˆ Y ˘ ` ż X W r r p ˆ π k x , π k x q ˆ µ k p dx q ` ε 1 k Ý Ñ 0 . In particular, 1 ´ ε 1 k ď 1 ´ ε k . 4.3 An alternativ e form of WSOT T o apply Theorem 2.13, it is conv enien t to work with the canonical em b edding J : J : P p X ˆ Y q Ñ P p X ˆ P p Y qq , π ÞÑ pro j X p π qp dx q δ π x p dp q . Consider its left-inv erse, the X ˆ Y -in tensity map ˆ I , and the intensit y map I : ˆ I : P p X ˆ P p Y qq Ñ P p X ˆ Y q , I : P p P p Y qq Ñ P p Y q , P ÞÑ ż p P P p Y q p p dy q P p dx, dp q , Q ÞÑ I p Q qp dy q : “ ż P p Y q p p dy q Q p dp q . W e further define the set Λ p µ, ν q : “ ! P P P p X ˆ P p Y qq : ˆ I p P q P Π p µ, ν q ) . Prop osition 4.3. ([ 13 , Pr op osition A.9]) L et g : Y Ñ r 1 , `8q b e c ontinuous, C : P g p Y q Ñ R b e c onvex, lower semic ontinuous and lower b ounde d by a ne gative multiple of ˆ g . Then for al l Q P P ˆ g p P p Y qq holds C p I p Q qq ď ż P g p Y q C p p q Q p dp q . No w we can in tro duce the alternativ e formulation (WSOT’): inf π P Λ S p µ,ν q ż R ˆ P p R q C p x, p q P p dx, dp q , where Λ S p µ, ν q is the set of all P P Λ p µ, ν q with full measure on tp x, p q P R ˆ P 1 : x ě ş R y p p dy qu . 26 4.4 Comp etitor and finite optimalit y T o formulate the monotonicity principle (Section 2.3), we recall the notions of comp etitor and finite optimalit y . Definition 4.4 (Comp etitor) . Let π b e a finite measure on R 2 with finite first moment, and let π 1 b e its first marginal. Fix a disintegration π “ π 1 ˆ κ x . Let M 0 , M 1 Ă R b e Borel. A measure π 1 is an p M 0 , M 1 q -c omp etitor of π if it has the same marginals as π and if, writing π 1 “ π 1 ˆ κ 1 x , one has ż R y κ 1 x p dy q ď ż R y κ x p dy q for π 1 -a.e. x P M 0 , ż R y κ 1 x p dy q “ ż R y κ x p dy q for π 1 -a.e. x P M 1 . Definition 4.5 (Finite optimality) . Let G : R ˆ R Ñ R b e a Borel cost function and let Γ Ă R 2 b e Borel. 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